\section{向量空间}

	\begin{titwo}
		已知 $\mathbb{R}^{3}$ 的两个基分别为
		\[
			\bm \alpha_{1} = \begin{bsmallmatrix}
				1 \\
				1 \\
				1
			\end{bsmallmatrix},
			\bm \alpha_{2} = \begin{bsmallmatrix}
				1 \\
				0 \\
				-1
			\end{bsmallmatrix},
			\bm \alpha_{3} = \begin{bsmallmatrix}
				1 \\
				0 \\
				1
			\end{bsmallmatrix}
		\]
		与
		\[
			\bm \beta_{1} = \begin{bsmallmatrix}
				1 \\
				2 \\
				1
			\end{bsmallmatrix},
			\bm \beta_{2} = \begin{bsmallmatrix}
				2 \\
				3 \\
				4
			\end{bsmallmatrix},
			\bm \beta_{3} = \begin{bsmallmatrix}
				3 \\
				4 \\
				3
			\end{bsmallmatrix},
		\]
		求由基 $\bm \alpha_{1},\bm \alpha_{2},\bm \alpha_{3}$ 到基 $\bm \beta_{1},\bm \beta_{2},\bm \beta_{3}$ 的过渡矩阵 $\bm P$.
	\end{titwo}

	\begin{titwo}
		设 $\bm \alpha_{1} = [1,0,1]^{\TT},\bm \alpha_{2} = [1,1,-1]^{\TT},\bm \alpha_{3} = [1,-1,1]^{\TT};$ $\bm \beta_{1} = [3,0,1]^{\TT}, \bm \beta_{2} = [2,0,0]^{\TT}, \bm \beta_{3} = [0,2,-2]^{\TT}$ 是 $\mathbb{R}^{3}$ 的两个基. 若向量 $\bm \xi$ 在基 $\bm \beta_{1},\bm \beta_{2},\bm \beta_{3}$ 下的坐标为 $[1,2,0]^{\TT}$，则 $\bm \xi$ 在基 $\bm \alpha_{1},\bm \alpha_{2},\bm \alpha_{3}$ 下的坐标为\kuo.

		\twoch{$[1,3,3]^{\TT}$}{$[-1,3,3]^{\TT}$}{$[-1,-3,3]^{\TT}$}{$[-1,3,-3]^{\TT}$}
	\end{titwo}

	\begin{titwo}
		设 $\mathbb{R}^{3}$ 中两个基
		\begin{gather*}
			\bm \alpha_{1} = [1,1,0]^{\TT}, \bm \alpha_{2} = [0,1,1]^{\TT}, \bm \alpha_{3} = [1,0,1]^{\TT}; \\
			\bm \beta_{1} = [1,0,0]^{\TT}, \bm \beta_{2} = [1,1,0]^{\TT}, \bm \beta_{3} = [1,1,1]^{\TT}.
		\end{gather*}
		\begin{enumerate}
			\item 求 $\bm \beta_{1},\bm \beta_{2},\bm \beta_{3}$ 到 $\bm \alpha_{1},\bm \alpha_{2},\bm \alpha_{3}$ 的过渡矩阵;
			\item 已知 $\bm \xi$ 在基 $\bm \beta_{1},\bm \beta_{2},\bm \beta_{3}$ 下的坐标为 $[1,0,2]^{\TT}$，求 $\bm \xi$ 在基 $\bm \alpha_{1},\bm \alpha_{2},\bm \alpha_{3}$ 下的坐标;
			\item 求在上述两个基下有相同坐标的向量.
		\end{enumerate}
	\end{titwo}